Method for figure control of optical surfaces

ABSTRACT

A method for figuring an optical surface of an optical element to achieve a target profile for the optical surface includes: applying a removal process to an extended region of the optical surface extending along a first direction to remove material from the extended region of the optical surface; adjusting a position of the optical surface relative to the removal process along a second direction perpendicular to the first direction to remove material from additional extended regions of the optical surface extending along the first direction at each of different positions of the optical surface along the second direction; and repeating the applying of the removal process and the adjusting of the optical surface relative to the removal process for each of multiple rotational orientations of the optical surface about a third direction perpendicular to the first and second directions to achieve the target profile of the optical surface.

CROSS REFERENCE TO RELATED APPLICATION

This application claims priority to U.S. Provisional Patent ApplicationNo. 62/812,663, filed on Mar. 1, 2019, pursuant to 35 USC § 119. Theentire content of this provisional application is herein incorporated byreference in its entirety.

BACKGROUND

The manufacturing of an optical element typically involves the shaping(or “figuring”) of at least on optical surface of the element relativeto a target profile for the optical surface. This target surface can bea plane, a sphere, or some other defined form such as an asphericsurface or even a free form surface. Techniques for achieving suchprofiles are varied and typically involve grinding and polishing overthe full aperture of the optical element. For higher precision optics,the techniques typically further involve deterministic local (or“sub-aperture”) processing of the optical surface to achieve the targetprofile.

Sub-aperture figuring involves the use of some small (relative to thesize of the substrate) removal function that dwells across the surfaceat varying speeds to locally remove material in sub regions of a largersubstrate to correct the global figure. There are many technologiesavailable to perform sub-aperture figure correction. Magneto-rheologicalFinishing (MRF) is a common commercially available technique. Ion BeamFiguring (IBF) is another method that is heavily leveraged due to thestability of the removal function. Most sub-aperture finishingtechniques are limited in some way by physical constraints (e.g.,pressures applied, tool wear, tool displacement tolerances, machinevelocity or acceleration constraints, etc . . . ) and are also typicallylimited in volumetric removal rate when used in any reasonably stableconditions due to the local nature of the process. This can lead to longprocessing times. Other limitations to this type of processing sometimesinclude, for example, degradation in the micro-roughness, issues withsurface quality, introduction of mid-spatial surface error, and edgeexclusion.

As a result, it is common for multiple processing techniques to be usedto figure a single optical element. For example, it is common to use afiguring technique in a larger ‘bulk’ removal mode to correct the longerspatial length surface error and then switch modes to use a smallerremoval function to address the higher spatial periods in a substrate.This is done to increase convergence efficiency and minimize the processtime.

In sum, the determinism of sub-aperture processing provides a greatadvantage to efficiently shape optical substrates. However, this cancome with trade-offs in surface quality, edge effects, roughness,run-time, or a degradation in the higher spatial frequencycharacterization of the surface. Such trade-offs can be increasinglyproblematic when a relatively large volume of material must be removedto achieve the desired results, as is generally the case for largeoptical elements.

SUMMARY

Disclosed herein are embodiments for a method for figuring opticalsurfaces. Embodiments address issues with conventional sub-aperturefinishing and still maintain a high level of determinism. They involve amix of global and local figuring approaches to provide removal rateefficiency comparable to full aperture processes and the determinism ofa sub-aperture processes.

In general, in one aspect, a method for figuring an optical surface ofan optical element to achieve a target profile for the optical surfaceis disclosed. The method includes: a) applying a removal process to anextended region of the optical surface extending along a first directionto remove material from the extended region of the optical surface; andb) adjusting a position of the optical surface relative to the removalprocess along a second direction perpendicular to the first direction toremove material from additional extended regions of the optical surfaceextending along the first direction at each of different positions ofthe optical surface along the second direction. During the applying ofthe removal process and the adjusting of the optical surface relative tothe removal process, the optical surface has a first rotationalorientation about a third direction perpendicular to the first andsecond directions. The method further includes: c) repeating theapplying of the removal process and the adjusting of the optical surfacerelative to the removal process for each of one or more additionalrotational orientations of the optical surface about the third directionto achieve the target profile of the optical surface.

Embodiments of the method may include one or more of the followingfeatures.

The extended region of the optical surface extending along the firstdirection from which material is removed by the removal process mayextend across a full aperture of the optical surface along the firstdirection. In certain embodiments, a full aperture of the opticalsurface is greater than 25 cm, or greater than 50 cm, or even greaterthan 100 cm.

The removal process is preferably laterally extended along the firstdirection to simultaneously and/or uniformly remove the material fromthe region of the optical surface extending along the first direction.

The removal process may be an etching process wherein at least part ofthe optical surface is immersed into an etching bath. For example, theadjusting of the optical surface relative to the removal process mayinclude immersing the optical surface into the etching bath along thesecond direction. Furthermore, immersing the optical surface into theetching bath along the second direction may include varying a speed ofthe immersion of the optical surface into the etching bath along thesecond direction to cause a rate at which the material at the additionalextended regions of the optical surface extending along the firstdirection are removed at each of the different positions of the opticalsurface along the second direction to vary nonlinearly with respect todistance along the second direction. In such cases, the extended regionsof the optical surface that are immersed in the etching bath may havematerial continuously removed by the etching bath in proportion to adwell time in the etching bath for each of the extended regions. Incertain embodiment, the method may further include adjusting atemperature of the etchant bath to adjust an etching rate for theremoval process.

In certain other embodiments, the removal process is an ion beam etchingprocess, wherein a dimension of the ion beam for the ion beam etchingprocess along the first direction is at least ten times greater than adimension of the ion beam along the second direction.

In certain other embodiments, the removal process is amagneto-rheological finishing processing implementing a cylindrical headto provide laterally extended removal process.

More generally, for any removal process, adjusting the position of theoptical surface relative to the removal process along the seconddirection may include varying a speed of the relative positioning alongthe second direction to cause a rate at which the material at theadditional extended regions of the optical surface extending along thefirst direction are removed at each of the different positions of theoptical surface along the second direction to vary nonlinearly withrespect to distance along the second direction.

The method may further include mounting the optical element in a fixtureto establish an initial rotational orientation of the optical surfaceabout the third direction prior to an initial application of the removalprocess to the optical surface. Furthermore, the method may furtherinclude reorienting the optical element in the fixture to establish eachof the one or more additional rotational orientations of the opticalsurface prior to repeating each of the corresponding applications of theremoval process and adjustments of the optical surface relative to theremoval process.

For example, the first rotational orientation and the one or moreadditional rotational orientations collectively include at least fourdifferent rotational orientations, eight different orientations, or evenmore than eight different orientations. In certain preferredembodiments, for example, the at least four different rotationalorientations may be selected from rotations corresponding to integermultiples of 45 degrees.

The method may further include expressing the target profile for thefiguring of the optical surface as a superposition of polynomialfunctions of a coordinate for the second dimension for each of the firstrotational orientation and the one or more additional rotationalorientations. For example, the adjustment of the optical surfacerelative to the removal process along the second direction for each ofthe first rotational orientation and the one or more additionalrotational orientations may be based on the polynomial function for eachcorresponding one of the first rotational orientation and the one ormore additional rotational orientations. Furthermore, a speed of theadjustment of the optical surface relative to the removal process alongthe second direction for each of the first rotational orientation andthe one or more additional rotational orientations may correspond to aderivative of the polynomial function with respect to the coordinate forthe second dimension for each corresponding one of the first rotationalorientation and the one or more additional rotational orientations.

In certain embodiments, a desired profile for the figuring of theoptical surface can be expressed in terms of coefficients for a set ofZernike polynomials, and the method may include approximating thedesired profile with the target profile expressed as the superpositionof polynomial functions for each of the first rotational orientation andthe one or more additional rotational orientations.

Embodiments of the method may include any of the following advantages:i) high ‘effective’ removal rates for large substrates; ii) batchprocessing for similar figure errors, thereby increasing throughput;iii) metrology limited wedge control and/or zero edge degradation; iv)profiling of free-form and aspheric geometries; v) profiling ofhigh-aspect ratio or mechanically fragile substrates; vi) processingthat is inherently free of subsurface damage and/or defect contamination(e.g., scratches/digs); vii) minimal roughness degradation on fusedsilica; and viii) simple implementation and high determinism, at lowcost and high-throughput.

The details of one or more embodiments of the invention are set forth inthe accompanying drawings and the description below. Other features,objects, and advantages of the invention will be apparent from thedescription and drawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIGS. 1A and 1B are schematic diagrams illustrating the method for twoangular orientations of the optical surface 112 being processed(0-degrees in FIG. 1A and 45-degrees in FIG. 1B).

FIG. 2 is a table list the first thirty-six Zernike polynomials.

FIG. 3 is a graphical representation in X-Y of the eight Zernikepolynomials up to fourth-order spherical.

FIG. 4 is graphical matrix of graphically illustrating how superpositionof one-dimension surface profiling at different angular orientations canproduce surface profiles corresponding to different Zernike polynomials.

FIG. 5 is an exemplary numerical matrix for the one-dimension correctionalong the Y-axis for each of eight 45-degree increment rotationsnecessary to correct for a surface error correction expressed in termsof the first sixteen Zernike polynomials (z₁, z₂, . . . z₁₆).

FIGS. 6A and 6B are images of the transmitted wavefront error (TWE) of afused silica window measured before (6A) and after (6B) profile etching.

FIG. 7 is a set of three different perspective images of a syntheticfringe overlay of a scaled version of an aspherical surface profilingdeparture applied to a plano-convex lens implemented by the methoddisclosed herein.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

Disclosed herein are methods for the correction of surface figure errorof an optical substrate in mixed mode between sub-aperture and fullaperture processing. This is done by utilizing a line-removal processthat provides a line-removal function that can continually scale toaccommodate up to the full-width of the substrate and repeating suchfull-width removal processes for different angularly orientations of thesubstrate to collectively achieve variable surface profiling along allnecessary directions. For example, the surface error correctionnecessary to achieve a target surface provide is decomposed into angularcomponents, and the optical surface is then processed with a collectiveset of full-width line-removal processes for respective angularorientations of the substrate that achieve the surface error correctionall those angular components.

In one embodiment, the line-removal process is a wet acid that willcontinually etch the portion of the substrate that is in contact withthe solution. In another embodiment, the line-removal process is anyprocess that provides a line contact with the optical surface, such asconventional polishing, MRF polishing, or IBF techniques that aremodified to be laterally extended and thereby provide a line-removalfunction. In certain embodiment, the decomposition of the surface errorcorrection into angular components is performed by expressing thesurface error correction in terms of coefficients for Zernike polynomial(as is conventionally done, although expression in terms of otherfunctions is also possible) and decomposing the Zernike polynomials intoa superposition of line functions for the different angular orientationsof the substrate, as is described in more detail below.

A line-removal process allows surface error over the full aperture ofthe optical surface to be corrected along the dimension corresponding tothe scan direction of the line-removal process (e.g., along the lengthof the surface), provided the removal function extends along the entirewidth of the aperture. One trivial example is that of a wedgecorrection. An extended removal function can readily impart (or remove)wedge. For example, by appropriately orienting the optical element suchthat the direction of the wedge is perpendicular to the removal function(which extends the entire width of the substrate) then the wedge can becontrolled by traversing the length of the substrate and dwelling longerwhere more removal is required. More generally, in other examples,varying this removal function along the length can create curves inadditional to a linear wedge (but only in the single-direction beingtraversed). For example, a trivial example of a curved figuring is thatof a cylinder, however, any sort of curve can be imparted, provided thatthe variation is perpendicular to the removal function. Repeating thislinear-removal process for different orientations of the substrateenables variable surface figuring along many directions.

Zernike polynomials are a fairly standard means to describe form erroron optical substrates and are functions of a radial component and anangular component. These polynomials begin by describing mean error,tilt, and power and then grow in complexity as the order of the termsbecome larger. It is common to use up to the first thirty-six todescribe a surface. Fitting even higher-order terms tends to losepractical value in many cases. For the purpose of the methods disclosedherein, however, what is important is that these polynomials can also besubstantially characterized by linear traces that recombine at discreteangles. By decomposing these polynomials, the first sixteen terms can bere-expressed by corresponding superpositions of linear traces at eightdiscrete angle 45-degrees apart. This is to say that Zernike polynomialscan be represented by a series of 1D curves (that extend the width ofthe substrate) at the eight different angular orientations. As a basicexample, tilt in orthogonal “X” and “Y” directions are just ‘curves’ attwo angular orientations 90-degrees apart. The same is true of power andastigmatism. These are just ‘cylinders’ added together with 90-degreeclocking (and in the astigmatic case the cylinders are inverted). Theprinciple likewise applies as the Zernike order grows in complexity(e.g., with respect to coma, 3^(rd) order spherical, etc . . . ),however, the linear combinations require clocking angles thatincreasingly become smaller (e.g., 45 degrees, 22.5 degrees, etc . . .).

The method is illustrated schematically in FIGS. 1A and 1B. An opticalelement 110 having an optical surface 112 whose surface figure is to becorrected to establish a target profile is mounted in an opticalmounting fixture 120 that orients the optical surface 112 relative to aline-removal process 140. In the Cartesian coordinate system illustratedin the FIGS. 1A and 1B, the optical surface 112 is nominally oriented inthe X-Y plane and the line-remove process extends along X-axis. Theline-remove process 140 uniformly removes material from the opticalsurface 112 across the full aperture of the optical surface along thedirection of the X-axis. The amount of material removed, and thecorresponding figuring, increases with the dwell time of theline-removal process 140 at the Y-axis coordinate for the line-removalprocess. To variably figure the optical surface 112 along theY-direction, the line-removal process 140 is scanned relative to theoptical surface 112 along the Y-direction with dwell times correspondingto the target profile.

To enable figuring along additional dimensions of the optical surface112, the line removal process 140 is repeated for each of one or moreadditional angular orientations of the optical surface 112 relative tothe line removal process 140. Specifically, the optical surface isrotated about a Z-axis perpendicular to the X- and Y-axes in theCartesian coordinate system shown in FIGS. 1A and 1B. For example, thiscan be accomplished by mounting elements 130 in fixture 120 thatrotatable support optical element 110 in fixture 120. Accordingly, FIG.1B illustrates the line-removal process 140 being applied to the opticalsurface 112 following a 45-degree rotation of the optical surface aboutthe Z-axis relative to the angular orientation of the optical surface inFIG. 1A. In other embodiments, the fixture 120 supporting opticalelement can be rotated as a whole relative to the line-removal process.

As mentioned above, in certain embodiments of the method, theline-removal process is a provided by an acid bath, and the method isillustrated below in greater detail with respect to this acid process.

Profile Etching

In describing this technique, it is appropriate to begin with thefundamental elements involved and build up to the ultimateimplementation. Consider the wet etching of an optical component forstock removal using an acid bath. In practice, the etch rate will bestable and isotropic such that, by fully submerging the substrate in theacid, a uniform removal of material will be performed. The change inheight (h) of the surface will be defined by Δh=ER·Δt, where ERrepresents the removal rate of the acid and Δt is the elapsed time.Technically, the etching described above qualifies as a deterministicfull aperture process in the most basic form of uniform removal but asyet, there is no basis for figure control. The scenario can be adjustedslightly by introducing the concept of a partially submerged substrate.By approximating a step function defined by the interface between theair and the acid bath, the system can be re-expressed by the followingequation (1).

$\begin{matrix}{{\Delta\;{h\left( {{\Delta\; t},y} \right)}} = \left\{ \begin{matrix}{{{{ER} \cdot \Delta}\; t},} & {y = {{in}\mspace{14mu}{acid}}} \\{0,} & {y = {{in}\mspace{14mu}{air}}}\end{matrix} \right.} & (1)\end{matrix}$

There is now a positional dependence of the removal defined by thelocation (y) of a point on the substrate and it becomes more appropriateto describe the system as having a removal function instead of a removalrate. Defining the setup this way makes clear that by introducingconventional kinematics into the system it is possible to obtaincontrolled non-uniformity of the etching in a similar fashion as thatemployed in many sub aperture processes. For example, an initialimplementation is that of a controlled immersion of a substrate into theacid at a fixed velocity in order to correct wedge.

Wedge is simply a linear amplitude gradient that exists about a givenlength (L) of a substrate. To correct a wedge, it is necessary topreferentially etch the thicker portion of the substrate by lowering thesubstrate into the acid bath in a controlled fashion. Since there isfreedom to choose the substrate orientation, the amplitude gradient isdefined to have a maximum deviation (Δh=h_(max)) at the bottom of thesubstrate and (Δh=0) at the top. From equation (1), the time (Δt)required to remove h_(max) is simply h_(max)/ER. Therefore, theimmersion velocity (v_(y)) is equal to L/Δt. Substituting for Δt andnoting that L/h_(max) can be defined as the inverse of the surface slope(m_(wedge)), the final velocity required to correct (or impart) thewedge is defined in equation (2).

$\begin{matrix}{v_{y} = \frac{ER}{m_{wedge}}} & (2)\end{matrix}$

There is an implicit requirement that the slope (m) must be positive(m>0), which would prove restricting when expanding the figurecorrection beyond wedge. However, there is an easy solution. Because theremoval is independent of any position (x) along the width, the rotationof the substrate by 180° serves to invert the target profile. Therefore,negative slopes (m<0) will become positive in what will be a mirroredversion of the profile. Thus m₁₈₀=−m and the correction again becomespossible. The importance of this relation will become apparent whenexpanding the concept to arbitrary profiles.

Now assume there exists a function h(y) that represents an arbitraryamplitude variation about y. Traversing along the profile h(y) by anarbitrary distance Δy will have an associated change in profileamplitude Δh, thereby the concept of wedge (Δh/Δy) is reintroduced on alocal scale, and a new velocity function v(y) can be established toperform the local correction. As the limit of Δy→0, the slope m(y) isthe derivative of the surface profile at location y, thus m(y)=dh/dy andv(y)=ER/m(y). Given the arbitrary nature of h(y), the sign of the slopewill also be arbitrary. As noted above, a single immersion process canonly correct positive slopes. However, as also noted above, a secondimmersion process with a rotation of the substrate by 180° will convertnegative slopes to positive ones. Thus, a complete change to the surfaceprofile is performed in rotated pairs and the final correction equationis as follows, where the subscripts dictate the separate 0° and 180°increments.

$\begin{matrix}{{v(y)}_{0,180} = \left\{ \begin{matrix}{\frac{ER}{\left( \frac{{dh}_{0,180}}{dy} \right)},} & {\left( \frac{{dh}_{0,180}}{dy} \right) > 0} \\{\infty,} & {\left( \frac{{dh}_{0,180}}{dy} \right) \leq 0}\end{matrix} \right.} & (3)\end{matrix}$

It should be noted that in the situation where the slopes are negativeand the velocity equates to infinity, there would be no removal intheory. In practice, the velocity is maximized to the constraints of thesystem and some minimal removal is unavoidable.

There now exists a means to impart any general profile h(y) onto thesurface(s) of an optical component using a wet acid bath. However, thedependence on y only is limiting, and it is demonstrated in thefollowing section that the removal function can be expanded beyond thislimitation. In fact, the concept of decomposing Zernike polynomials(which are functions of x and y) into angular components that aredependent on y and immersion angle θ is introduced, and it will be shownthat the acid etch process can be used to recreate these polynomials.

Zernike Decomposition

As was noted above, Zernike polynomials are a fairly standard means todescribe form error on optical substrates. They are most familiarlyrecognized by the descriptions of the early terms (power, coma,astigmatism, trefoil, etc . . . ) but consist of an infinite series oforthogonal functions of increasing complexity. FIG. 2 is a tableproviding the first thirty-six Zernike polynomials in terms of theradial component “ρ” and an angular component “∅”. As is readilyapparent, the expressions of these polynomials become cumbersome whenfully expanded, and so will the expressions associated with thedecomposition into angular components. Therefore, it will be practicalto switch to a graphical representation and it will be understood thatthe descriptions that follow can be demonstrated (and proven)computationally. FIG. 3 is a graphical representation of eight Zernikepolynomials up to 4^(th) order spherical.

In order to recreate the first series of Zernike profiles shown in FIG.3 using the etching method described above, the wet etch immersionprocess is repeated as varying discrete angles. To correct for the nineZernike polynomials illustrated in FIG. 3, the angles are confined to45° increments. Of course, in other embodiments and/or for correctingeven higher-order Zernike polynomials smaller angle increments can beused. Additionally, since profiling is done in pairs (0° and 180°apart), reference to future immersion angles will be described by thefirst orientation only and the repeated 180° etch will be implied.Therefore, the immersion angles will be designated as θ_(i) and i willassume the values 0, 45, 90, 135 . . . 315.

FIG. 4 graphically illustrates how one-dimension profiling at each ofseveral of these different immersion angles can be combined insuperposition to create a specific profiling multi-dimensional profilingcorresponding to a Zernike polynomial. Specifically, the images on thetop left in FIG. 4 demonstrate how the Zernike polynomials can be splitinto angular components that are merely functions of one variable (y) ifproperly oriented, and can therefore be corrected using the etchprofiling method. Each individual immersion will act to linearlyrecombine the individual profiles to recreate the appropriate Zernike(the summation of each profile from left to right in FIG. 4). In thecases shown for power, astigmatism, coma, and 3^(rd) order spherical, acomplete recreation can be performed by as few as 4 discrete angles. Itcan be shown that the first 16 terms can be recreated as above in as fewas 8 rotations. Higher order terms require more rotations.

It is not necessary that each individual polynomial used to describe thesurface be imparted sequentially. Each discrete angle will havecontributions from each Zernike term and can be profiled concurrentlybased upon the sums of the profiles at each angle (the summation of eachprofile from top to bottom in FIG. 4). Returning back to themathematical model, a slight modification can be made to equation (3) toinclude the angular rotation and a new function Z_(n)(y) is created foreach immersion angle (θ), where Z_(n) represents the angular profileassociated with the n^(th) Zernike term. The complete picture isrepresented in equations (4) and (5) where θ will assume the values0,45, . . . ,315.

$\begin{matrix}{{h(y)}_{\theta} = {\sum\limits_{n = 1}^{n = 16}{Z(y)}_{n,\theta}}} & (4) \\{{v(y)}_{\theta} = \left\{ \begin{matrix}{\frac{ER}{\left( \frac{{{dh}(y)}_{\theta}}{dy} \right)},} & {\left( \frac{{{dh}(y)}_{\theta}}{dy} \right) > 0} \\{\infty,} & {\ {\left( \frac{{{dh}(y)}_{\theta}}{dy} \right) \leq 0}}\end{matrix} \right.} & (5)\end{matrix}$

By way of example only, a matrix representation for the one-dimensioncorrection along the Y-axis for the eight 45-degree increment rotationsnecessary to correct for a surface error correction expressed in termsof the first sixteen Zernike polynomials (z₁, z₂, . . . z₁₆) is shown inFIG. 5. The lower-order terms require only one discrete orientationwhile the amount of clocking required increases as the terms go tohigher order. The matrix shows the Zernike terms increasing in order byrow and the discrete angular breakdown along the columns (at 45 degreeincrements). The decoupling of the equations is not necessarily uniqueand the values populating the matrix are but one possible solution.

EXAMPLES

In this section, examples of two components that have been processedwill be described. Additionally, there are some unique advantages of theprocess that will be highlighted concerning optical windowmanufacturing, minimization of edge effects, aspherization, scalabilitytowards large substrates, and bulk processing. In regards to thepractical process implementation, the optics referenced below were fusedsilica (SiO2) and etched using a buffered hydrofluoric acid solution(BOE).

In the first example, the transmitted wavefront error (TWE) of a fusedsilica window was measured before and after profile etching. Acomparison was made between the Zernike coefficients used to describethe wavefront error and a very high convergence was achieved and isdemonstrated in FIGS. 6A (before) and 6B (after) and Table 1 below.

TABLE 1 Zernike Description Initial (waves) Post-Etch (waves) Tilt X−0.069 0.000 Tilt Y 0.275 0.001 Focus −0.151 0.000 Astig 0, 90 0.0170.001 Astig +− 45 −0.014 0.012 X Coma 0.005 −0.003 Y Coma −0.008 0.005Sphere 0.010 0.005

Because the component was an optical window and the requirement wasspecified in TWE, both surfaces could be processed at the same time. Thedouble sided application of the process highlights a potential advantagein a production environment since this will effectively double theremoval rate of the correction. Another benefit is shown in the fidelityof the data out to the very edge of the substrate. The edge effect fromprocessing is too small to be quantified from the associated data takenand this demonstrates another advantage of the process.

In the second example, the process was utilized to aspherize the surfaceof a curved geometry as oppose to correcting the figure error of theplanar geometry of a window. The aspheric departure was characterized bythe conic (k) only and the deviation was completely described by Zernikepolynomials up to 3^(rd) order spherical. For visualization purposes, asynthetic fringe overlay of a scaled version of the aspheric departureis shown in FIG. 7. In this application, the process was implemented forbulk convergence towards the final surface profile and used in acomplementary fashion with other deterministic finishing techniques tofully fabricate the optic. Two significant advantages of the process arehighlighted in this example. The first is the demonstration of theindependence of the etching on the surface geometry. The acid willalways contour to the geometry of the surface and by (mathematically)projecting the desired profile onto a virtual plane, the process can bereadily applied. The second is a less apparent benefit from thedouble-sided nature of the process. Since the etching was performedsimultaneously on both surfaces of the optic, the profile change on theplanar surface was equal to the change on the curved geometry of theaspheric surface. In this situation, it was possible to leverageexisting interferometric capability for planar surfaces in order tocharacterize the opposing surface profile where the equivalentcapability was unavailable.

So far, there has been little reference to removal rates of the process.In part, this is due to the fact that the etch rates can varydramatically based on concentration and temperature. However, there isalso the notion of an effective removal rate in the etch process thatmust be highlighted. Although the etch rate is stable at a fixedtemperature (and concentration), the ultimate volumetric removal rate onthe substrate will vary with the size of the substrate and this issubtly implied from equation (2) regarding wedge correction. It can benoted that, for a given amplitude of correction, if the length of thesubstrate gets larger, then the velocity will scale proportionately. Inother words, the processing time is the same independent of the size ofthe substrate and the volumetric removal rate of the system is not aconstant.

As an example, consider a typical etch rate of ˜80 nm/min and a 100 mmdiameter circular substrate. In order to correct 1 um of wedge theassociated velocity will be 8 mm/min and the correction will take 12.5minutes. If the diameter is increased to 1000 mm, the velocity goes to80 mm/min and the correction time remains the same. Of course, theamount of material removed has increased by a factor of a hundred andthe effective removal rate jumps from ˜0.3 mm³/min to ˜30 mm³/min. Thus,one of the primary advantages of the etch profiling technique isexemplified by the drastic increase in removal efficiency as thesubstrates grow in size.

The acid profiling approach has proven to be very useful forestablishing an efficient deterministic process that satisfies theprocess time (high removal rate) and geometry restraints involved inoptics manufacturing. The stability of the acid etch rate contributes tothe high level of determinism and provides some freedom of use for manydifferent sizes and geometries of a substrate. Combined with thekinematics of orientation and velocity, this mix between a full and subaperture approach has also demonstrated some desirable volumetricremoval rate efficiencies, especially when applied to increasinglylarger substrates.

Further Examples and Variations

The specific examples above have involved processing a single opticalsubstrate. However, it is also possible to carry out bulk processing ofmore than one substrate, especially where the substrates have likegeometries and the desired removal is consistent across each substrate.All this required is appropriate conventional tooling.

The specific examples above were limited to polished substrates nearingcompletion. However, there are many processes and technologies involvedin the manufacturing of an optical substrate and the acid profilingtechnique can also be used in a complementary fashion to support tosupport such other processes and technologies. For example, the methodsdisclosed herein can be used for bulk shaping in the early fabricationstages of an optical element and also to fabricate optical elements withcomplex freeform surfaces to leverage the geometry independence of themethods.

As noted above, the methods are not limited to an etch bath for theline-removal process. In other variations, the line-removal process isany process that provides a line contact with the optical surface, suchas conventional polishing, MRF polishing, or IBF techniques that aremodified to be laterally extended and thereby provide a line-removalfunction. For example, an ion beam source for IBF can be configured toprovide a line profile instead of a spot profile (e.g., for onedimension is at least ten times larger than an orthogonal dimension).Similarly, for example, a spherical head for MRF can be replaced with acylindrical head.

Furthermore, when using an etch bath for the line-removal process,variations including varying the etch rate of the bath during each ofone or more of the immersion processes and/or as between different onesof the ones of the immersion processes. This can be done, for example,by varying the temperature of the etch bath to vary its etch rate and/orvarying the composition of the etch bath to vary the etch rate.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of anyinventions or of what may be claimed, but rather as descriptions offeatures specific to particular embodiments of particular inventions.

Certain features that are described in this specification in the contextof separate embodiments can also be implemented in combination in asingle embodiment. Conversely, various features that are described inthe context of a single embodiment can also be implemented in multipleembodiments separately or in any suitable subcombination.

Moreover, although features may be described above as acting in certaincombinations and even initially claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.Accordingly, other embodiments are within the scope of the followingclaims.

What is claimed is:
 1. A method for figuring an optical surface of anoptical element to achieve a target profile for the optical surface, themethod comprising: a. applying a removal process to an extended regionof the optical surface extending along a first direction to removematerial from the extended region of the optical surface; and b.adjusting a position of the optical surface relative to the removalprocess along a second direction perpendicular to the first direction toremove material from additional extended regions of the optical surfaceextending along the first direction at each of different positions ofthe optical surface along the second direction, c. wherein, during theapplying of the removal process and the adjusting of the optical surfacerelative to the removal process, the optical surface has a firstrotational orientation about a third direction perpendicular to thefirst and second directions, and d. wherein the method furthercomprises: repeating the applying of the removal process and theadjusting of the optical surface relative to the removal process foreach of one or more additional rotational orientations of the opticalsurface about the third direction to achieve the target profile of theoptical surface, wherein the removal process is an etching processwherein at least part of the optical surface is immersed into an etchingbath, wherein the adjusting of the optical surface relative to theremoval process comprises immersing the optical surface into the etchingbath along the second direction, wherein immersing the optical surfaceinto the etching bath along the second direction comprises varying aspeed of the immersion of the optical surface into the etching bathalong the second direction to cause a rate at which the material at theadditional extended regions of the optical surface extending along thefirst direction are removed at each of the different positions of theoptical surface along the second direction to vary nonlinearly withrespect to distance along the second direction.
 2. The method of claim1, wherein the extended regions of the optical surface that are immersedin the etching bath have material continuously removed by the etchingbath in proportion to a dwell time in the etching bath for each of theextended regions.
 3. The method of claim 1, wherein the target profileof the optical surface comprises any of a wedge surface, a sphericalsurface, an aspherical surface, and a free-form surface.
 4. A method forfiguring an optical surface of an optical element to achieve a targetprofile for the optical surface, the method comprising: a. applying aremoval process to an extended region of the optical surface extendingalong a first direction to remove material from the extended region ofthe optical surface; and b. adjusting a position of the optical surfacerelative to the removal process along a second direction perpendicularto the first direction to remove material from additional extendedregions of the optical surface extending along the first direction ateach of different positions of the optical surface along the seconddirection, c. wherein, during the applying of the removal process andthe adjusting of the optical surface relative to the removal process,the optical surface has a first rotational orientation about a thirddirection perpendicular to the first and second directions, and d.wherein the method further comprises: repeating the applying of theremoval process and the adjusting of the optical surface relative to theremoval process for each of one or more additional rotationalorientations of the optical surface about the third direction to achievethe target profile of the optical surface, and further comprisingexpressing the target profile for the figuring of the optical surface asa superposition of polynomial functions of a coordinate for the seconddimension for each of the first rotational orientation and the one ormore additional rotational orientations.
 5. The method of claim 4,wherein the extended region of the optical surface extending along thefirst direction from which material is removed by the removal processextends across a full aperture of the optical surface along the firstdirection.
 6. The method of claim 4, wherein the removal process is anetching process wherein at least part of the optical surface is immersedinto an etching bath.
 7. The method of claim 6, wherein the adjusting ofthe optical surface relative to the removal process comprises immersingthe optical surface into the etching bath along the second direction. 8.The method of claim 6, wherein the method further comprising adjusting atemperature of the etchant bath to adjust an etching rate for theremoval process.
 9. The method of claim 4, wherein adjusting theposition of the optical surface relative to the removal process alongthe second direction comprises varying a speed of the relativepositioning along the second direction to cause a rate at which thematerial at the additional extended regions of the optical surfaceextending along the first direction are removed at each of the differentpositions of the optical surface along the second direction to varynonlinearly with respect to distance along the second direction.
 10. Themethod of claim 4, further comprising mounting the optical element in afixture to establish an initial rotational orientation of the opticalsurface about the third direction prior to an initial application of theremoval process to the optical surface.
 11. The method of claim 10,further comprising reorienting the optical element in the fixture toestablish each of the one or more additional rotational orientations ofthe optical surface prior to repeating each of the correspondingapplications of the removal process and adjustments of the opticalsurface relative to the removal process.
 12. The method of claim 4,wherein the first rotational orientation and the one or more additionalrotational orientations collectively comprise at least four differentrotational orientations.
 13. The method of claim 12, wherein the atleast four different rotational orientations comprise eight differentrotational orientations.
 14. The method of claim 12, wherein the atleast four different rotational orientations are selected from rotationscorresponding to integer multiples of 45 degrees.
 15. The method ofclaim 4, wherein the adjustment of the optical surface relative to theremoval process along the second direction for each of the firstrotational orientation and the one or more additional rotationalorientations is based on the polynomial function for each correspondingone of the first rotational orientation and the one or more additionalrotational orientations.
 16. The method of claim 15, wherein a speed ofthe adjustment of the optical surface relative to the removal processalong the second direction for each of the first rotational orientationand the one or more additional rotational orientations corresponds to aderivative of the polynomial function with respect to the coordinate forthe second dimension for each corresponding one of the first rotationalorientation and the one or more additional rotational orientations. 17.The method of claim 4, wherein a desired profile for the figuring of theoptical surface can be expressed in terms of coefficients for a set ofZernike polynomials, and where the method comprises approximating thedesired profile with the target profile expressed as the superpositionof polynomial functions for each of the first rotational orientation andthe one or more additional rotational orientations.
 18. The method ofclaim 4, wherein a full aperture of the optical surface is greater than25 cm.
 19. The method of claim 4, wherein a full aperture of the opticalsurface is greater than 50 cm.
 20. The method of claim 4, wherein a fullaperture of the optical surface is greater than 100 cm.
 21. The methodof claim 4, wherein the removal process is laterally extended along thefirst direction to simultaneously remove the material from the region ofthe optical surface extending along the first direction.
 22. The methodof claim 4, wherein the removal process is laterally extended along thefirst direction to uniformly remove the material from the region of theoptical surface extending along the first direction.
 23. The method ofclaim 4, wherein the removal process is an ion beam etching process. 24.The method of claim 23, wherein a dimension of the ion beam for the ionbeam etching process along the first direction is at least ten timesgreater than a dimension of the ion beam along the second direction.